Could you help me solve a physics problem? An axis of length L and mass M is balanced on the side of the table at its center of gravity. A ball with a relative height h is dropped onto its most protruding end. What is the speed of the ball immediately after impact? Sure! To solve this problem, we can use the law of conservation of energy. The initial potential energy of the ball is equal to its final kinetic energy after impact.

The initial potential energy is given by:

U = mgh, where m is the mass of the ball, g is the acceleration due to gravity, and h is the relative height.

The final kinetic energy is given by:

K = (1/2)mv^2, where v is the speed of the ball immediately after impact.

Setting U = K, we have:

mgh = (1/2)mv^2

Solving for v, we get:

v = sqrt(2gh)

So the speed of the ball immediately after impact is equal to the square root of 2 times the acceleration due to gravity times the relative height of the ball. Does that mean that the length L and mass M of the axis have no affect on the speed of the ball after impact?
The edge of a table is a fulcrum for the axis object with L and mass M, and the axis object is a lever. 

Under certain assumptions such as zero fulcrum friction and axis object mass of 0, the ball will continue to go the same speed.

If the ball drops onto the lever and effort force is closer to the fulcrum than the load of the axis object (lever), you are working at a mechanical disadvantage and the ball will slow down due to torque (Work = Force * Distance).
Impacts are complex physics problems. Amusing the ball has mass, it cannot change velocity instantly, so its speed just before and just after the impact must be the same.

At the exact moment of impact, the ball will be compressed, so the bottom side of the ball will slow down before the top of the ball slows down. There won't be a single speed that describes all parts of the ball. The change in velocity will travel from the bottom of the ball to the top of the ball at the speed of sound in the ball material. The axis will experience a similar deformation. Depending on the material properties, the material may also permanently deform at the contact point, taking energy away from the velocity.

Ignoring the contact situation and assuming the axis and ball will move at the same velocity together, without the ball bouncing, the velocity after impact will depend on the mass and inertia of the axis, which can be calculated from it's length, and the ratio between the mass of the axis and the ball, which is unknown since the mass of the ball was not given.